I have a question about the lattice reduction algorithms like LLL algorithm.
Lattice reduction algorithms like LLL generate a unimodular matrix which makes more orthogonal basises for a given matrix. For example if B is a given matrix the LLL algorithm generates the matrix $\tilde{\mathbf B} = {\mathbf B}{\mathbf T}$ where $\tilde{\mathbf B}$ is a more orthogonal matrix and ${\mathbf T}$ is a unimodular matrix.
Indeed, the unimodular matrix ${\mathbf T}$ generated by the LLL algorithm may contain a zero or zeros.
For a specific application the matrix ${\mathbf T}$ should not contain any zero. Do you know any reduction algorithm with a non zero constraint for the matrix ${\mathbf T}$? (I mean that all the elements of the matrix ${\mathbf T}$ have to be non-zero.)